The rare decay $K_{\mathrm{L}} \rightarrow\pi^{0} \mu^{ + } \mu^{-}$ within the SM

We present an updated analysis of the rare decay \(K_{\mathrm{L}} \rightarrow\pi^{0} \mu^{ + } \mu^{-}\) within the standard model. In particular, we present the first complete calculation of the two-photon CP-conserving amplitude within chiral perturbation theory, at the lowest non-trivial order. Our results confirm the previous findings that the CP-conserving contribution to the decay rate cannot be neglected. By means of an explicit two-loop calculation, we show that this contribution can be estimated with sufficient accuracy compared to the CP-violating terms. We predict \({\cal B}(K_{\mathrm{L}}\rightarrow\pi^{0}\mu^{ + } \mu^{-})_{\mathrm{SM}} = (1.5\pm0.3)\times10^{-11}\), with approximately equal contributions from the CP-conserving component, the indirect-CP-violating term, and the interesting direct-CP-violating amplitude. The error of this prediction is mainly of parametric nature and could be substantially reduced with better data on the \(K_{\mathrm{S}}\rightarrow\pi^{0}\ell^{ + }\ell^{-}\) modes. The standard model predictions for various differential distributions and the sensitivity to possible new-physics effects are also briefly discussed.